Your data matches 233 different statistics following compositions of up to 3 maps.
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Matching statistic: St000897
St000897: Integer partitions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1]
=> 1 = 2 - 1
[2]
=> 1 = 2 - 1
[1,1]
=> 1 = 2 - 1
[3]
=> 1 = 2 - 1
[2,1]
=> 1 = 2 - 1
[1,1,1]
=> 1 = 2 - 1
[4]
=> 1 = 2 - 1
[3,1]
=> 1 = 2 - 1
[2,2]
=> 1 = 2 - 1
[2,1,1]
=> 2 = 3 - 1
[1,1,1,1]
=> 1 = 2 - 1
[5]
=> 1 = 2 - 1
[4,1]
=> 1 = 2 - 1
[3,2]
=> 1 = 2 - 1
[3,1,1]
=> 2 = 3 - 1
[2,2,1]
=> 2 = 3 - 1
[2,1,1,1]
=> 2 = 3 - 1
[1,1,1,1,1]
=> 1 = 2 - 1
[6]
=> 1 = 2 - 1
[5,1]
=> 1 = 2 - 1
[4,2]
=> 1 = 2 - 1
[4,1,1]
=> 2 = 3 - 1
[3,3]
=> 1 = 2 - 1
[3,2,1]
=> 1 = 2 - 1
[3,1,1,1]
=> 2 = 3 - 1
[2,2,2]
=> 1 = 2 - 1
[2,2,1,1]
=> 1 = 2 - 1
[2,1,1,1,1]
=> 2 = 3 - 1
[1,1,1,1,1,1]
=> 1 = 2 - 1
[7]
=> 1 = 2 - 1
[6,1]
=> 1 = 2 - 1
[5,2]
=> 1 = 2 - 1
[5,1,1]
=> 2 = 3 - 1
[4,3]
=> 1 = 2 - 1
[4,2,1]
=> 1 = 2 - 1
[4,1,1,1]
=> 2 = 3 - 1
[3,3,1]
=> 2 = 3 - 1
[3,2,2]
=> 2 = 3 - 1
[3,2,1,1]
=> 2 = 3 - 1
[3,1,1,1,1]
=> 2 = 3 - 1
[2,2,2,1]
=> 2 = 3 - 1
[2,2,1,1,1]
=> 2 = 3 - 1
[2,1,1,1,1,1]
=> 2 = 3 - 1
[1,1,1,1,1,1,1]
=> 1 = 2 - 1
[8]
=> 1 = 2 - 1
[7,1]
=> 1 = 2 - 1
[6,2]
=> 1 = 2 - 1
[6,1,1]
=> 2 = 3 - 1
[5,3]
=> 1 = 2 - 1
[5,2,1]
=> 1 = 2 - 1
Description
The number of different multiplicities of parts of an integer partition.
Matching statistic: St000159
Mp00095: Integer partitions to binary wordBinary words
Mp00178: Binary words to compositionInteger compositions
Mp00040: Integer compositions to partitionInteger partitions
St000159: Integer partitions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1]
=> 10 => [1,2] => [2,1]
=> 2
[2]
=> 100 => [1,3] => [3,1]
=> 2
[1,1]
=> 110 => [1,1,2] => [2,1,1]
=> 2
[3]
=> 1000 => [1,4] => [4,1]
=> 2
[2,1]
=> 1010 => [1,2,2] => [2,2,1]
=> 2
[1,1,1]
=> 1110 => [1,1,1,2] => [2,1,1,1]
=> 2
[4]
=> 10000 => [1,5] => [5,1]
=> 2
[3,1]
=> 10010 => [1,3,2] => [3,2,1]
=> 3
[2,2]
=> 1100 => [1,1,3] => [3,1,1]
=> 2
[2,1,1]
=> 10110 => [1,2,1,2] => [2,2,1,1]
=> 2
[1,1,1,1]
=> 11110 => [1,1,1,1,2] => [2,1,1,1,1]
=> 2
[5]
=> 100000 => [1,6] => [6,1]
=> 2
[4,1]
=> 100010 => [1,4,2] => [4,2,1]
=> 3
[3,2]
=> 10100 => [1,2,3] => [3,2,1]
=> 3
[3,1,1]
=> 100110 => [1,3,1,2] => [3,2,1,1]
=> 3
[2,2,1]
=> 11010 => [1,1,2,2] => [2,2,1,1]
=> 2
[2,1,1,1]
=> 101110 => [1,2,1,1,2] => [2,2,1,1,1]
=> 2
[1,1,1,1,1]
=> 111110 => [1,1,1,1,1,2] => [2,1,1,1,1,1]
=> 2
[6]
=> 1000000 => [1,7] => [7,1]
=> 2
[5,1]
=> 1000010 => [1,5,2] => [5,2,1]
=> 3
[4,2]
=> 100100 => [1,3,3] => [3,3,1]
=> 2
[4,1,1]
=> 1000110 => [1,4,1,2] => [4,2,1,1]
=> 3
[3,3]
=> 11000 => [1,1,4] => [4,1,1]
=> 2
[3,2,1]
=> 101010 => [1,2,2,2] => [2,2,2,1]
=> 2
[3,1,1,1]
=> 1001110 => [1,3,1,1,2] => [3,2,1,1,1]
=> 3
[2,2,2]
=> 11100 => [1,1,1,3] => [3,1,1,1]
=> 2
[2,2,1,1]
=> 110110 => [1,1,2,1,2] => [2,2,1,1,1]
=> 2
[2,1,1,1,1]
=> 1011110 => [1,2,1,1,1,2] => [2,2,1,1,1,1]
=> 2
[1,1,1,1,1,1]
=> 1111110 => [1,1,1,1,1,1,2] => [2,1,1,1,1,1,1]
=> 2
[7]
=> 10000000 => [1,8] => [8,1]
=> 2
[6,1]
=> 10000010 => [1,6,2] => [6,2,1]
=> 3
[5,2]
=> 1000100 => [1,4,3] => [4,3,1]
=> 3
[5,1,1]
=> 10000110 => [1,5,1,2] => [5,2,1,1]
=> 3
[4,3]
=> 101000 => [1,2,4] => [4,2,1]
=> 3
[4,2,1]
=> 1001010 => [1,3,2,2] => [3,2,2,1]
=> 3
[4,1,1,1]
=> 10001110 => [1,4,1,1,2] => [4,2,1,1,1]
=> 3
[3,3,1]
=> 110010 => [1,1,3,2] => [3,2,1,1]
=> 3
[3,2,2]
=> 101100 => [1,2,1,3] => [3,2,1,1]
=> 3
[3,2,1,1]
=> 1010110 => [1,2,2,1,2] => [2,2,2,1,1]
=> 2
[3,1,1,1,1]
=> 10011110 => [1,3,1,1,1,2] => [3,2,1,1,1,1]
=> 3
[2,2,2,1]
=> 111010 => [1,1,1,2,2] => [2,2,1,1,1]
=> 2
[2,2,1,1,1]
=> 1101110 => [1,1,2,1,1,2] => [2,2,1,1,1,1]
=> 2
[2,1,1,1,1,1]
=> 10111110 => [1,2,1,1,1,1,2] => [2,2,1,1,1,1,1]
=> 2
[1,1,1,1,1,1,1]
=> 11111110 => [1,1,1,1,1,1,1,2] => [2,1,1,1,1,1,1,1]
=> 2
[8]
=> 100000000 => [1,9] => [9,1]
=> 2
[7,1]
=> 100000010 => [1,7,2] => [7,2,1]
=> 3
[6,2]
=> 10000100 => [1,5,3] => [5,3,1]
=> 3
[6,1,1]
=> 100000110 => [1,6,1,2] => [6,2,1,1]
=> 3
[5,3]
=> 1001000 => [1,3,4] => [4,3,1]
=> 3
[5,2,1]
=> 10001010 => [1,4,2,2] => [4,2,2,1]
=> 3
Description
The number of distinct parts of the integer partition. This statistic is also the number of removeable cells of the partition, and the number of valleys of the Dyck path tracing the shape of the partition.
Mp00042: Integer partitions initial tableauStandard tableaux
Mp00207: Standard tableaux horizontal strip sizesInteger compositions
Mp00133: Integer compositions delta morphismInteger compositions
St000903: Integer compositions ⟶ ℤResult quality: 99% values known / values provided: 99%distinct values known / distinct values provided: 100%
Values
[1]
=> [[1]]
=> [1] => [1] => 1 = 2 - 1
[2]
=> [[1,2]]
=> [2] => [1] => 1 = 2 - 1
[1,1]
=> [[1],[2]]
=> [1,1] => [2] => 1 = 2 - 1
[3]
=> [[1,2,3]]
=> [3] => [1] => 1 = 2 - 1
[2,1]
=> [[1,2],[3]]
=> [2,1] => [1,1] => 1 = 2 - 1
[1,1,1]
=> [[1],[2],[3]]
=> [1,1,1] => [3] => 1 = 2 - 1
[4]
=> [[1,2,3,4]]
=> [4] => [1] => 1 = 2 - 1
[3,1]
=> [[1,2,3],[4]]
=> [3,1] => [1,1] => 1 = 2 - 1
[2,2]
=> [[1,2],[3,4]]
=> [2,2] => [2] => 1 = 2 - 1
[2,1,1]
=> [[1,2],[3],[4]]
=> [2,1,1] => [1,2] => 2 = 3 - 1
[1,1,1,1]
=> [[1],[2],[3],[4]]
=> [1,1,1,1] => [4] => 1 = 2 - 1
[5]
=> [[1,2,3,4,5]]
=> [5] => [1] => 1 = 2 - 1
[4,1]
=> [[1,2,3,4],[5]]
=> [4,1] => [1,1] => 1 = 2 - 1
[3,2]
=> [[1,2,3],[4,5]]
=> [3,2] => [1,1] => 1 = 2 - 1
[3,1,1]
=> [[1,2,3],[4],[5]]
=> [3,1,1] => [1,2] => 2 = 3 - 1
[2,2,1]
=> [[1,2],[3,4],[5]]
=> [2,2,1] => [2,1] => 2 = 3 - 1
[2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [2,1,1,1] => [1,3] => 2 = 3 - 1
[1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [1,1,1,1,1] => [5] => 1 = 2 - 1
[6]
=> [[1,2,3,4,5,6]]
=> [6] => [1] => 1 = 2 - 1
[5,1]
=> [[1,2,3,4,5],[6]]
=> [5,1] => [1,1] => 1 = 2 - 1
[4,2]
=> [[1,2,3,4],[5,6]]
=> [4,2] => [1,1] => 1 = 2 - 1
[4,1,1]
=> [[1,2,3,4],[5],[6]]
=> [4,1,1] => [1,2] => 2 = 3 - 1
[3,3]
=> [[1,2,3],[4,5,6]]
=> [3,3] => [2] => 1 = 2 - 1
[3,2,1]
=> [[1,2,3],[4,5],[6]]
=> [3,2,1] => [1,1,1] => 1 = 2 - 1
[3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> [3,1,1,1] => [1,3] => 2 = 3 - 1
[2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [2,2,2] => [3] => 1 = 2 - 1
[2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> [2,2,1,1] => [2,2] => 1 = 2 - 1
[2,1,1,1,1]
=> [[1,2],[3],[4],[5],[6]]
=> [2,1,1,1,1] => [1,4] => 2 = 3 - 1
[1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> [1,1,1,1,1,1] => [6] => 1 = 2 - 1
[7]
=> [[1,2,3,4,5,6,7]]
=> [7] => [1] => 1 = 2 - 1
[6,1]
=> [[1,2,3,4,5,6],[7]]
=> [6,1] => [1,1] => 1 = 2 - 1
[5,2]
=> [[1,2,3,4,5],[6,7]]
=> [5,2] => [1,1] => 1 = 2 - 1
[5,1,1]
=> [[1,2,3,4,5],[6],[7]]
=> [5,1,1] => [1,2] => 2 = 3 - 1
[4,3]
=> [[1,2,3,4],[5,6,7]]
=> [4,3] => [1,1] => 1 = 2 - 1
[4,2,1]
=> [[1,2,3,4],[5,6],[7]]
=> [4,2,1] => [1,1,1] => 1 = 2 - 1
[4,1,1,1]
=> [[1,2,3,4],[5],[6],[7]]
=> [4,1,1,1] => [1,3] => 2 = 3 - 1
[3,3,1]
=> [[1,2,3],[4,5,6],[7]]
=> [3,3,1] => [2,1] => 2 = 3 - 1
[3,2,2]
=> [[1,2,3],[4,5],[6,7]]
=> [3,2,2] => [1,2] => 2 = 3 - 1
[3,2,1,1]
=> [[1,2,3],[4,5],[6],[7]]
=> [3,2,1,1] => [1,1,2] => 2 = 3 - 1
[3,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7]]
=> [3,1,1,1,1] => [1,4] => 2 = 3 - 1
[2,2,2,1]
=> [[1,2],[3,4],[5,6],[7]]
=> [2,2,2,1] => [3,1] => 2 = 3 - 1
[2,2,1,1,1]
=> [[1,2],[3,4],[5],[6],[7]]
=> [2,2,1,1,1] => [2,3] => 2 = 3 - 1
[2,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7]]
=> [2,1,1,1,1,1] => [1,5] => 2 = 3 - 1
[1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> [1,1,1,1,1,1,1] => [7] => 1 = 2 - 1
[8]
=> [[1,2,3,4,5,6,7,8]]
=> [8] => [1] => 1 = 2 - 1
[7,1]
=> [[1,2,3,4,5,6,7],[8]]
=> [7,1] => [1,1] => 1 = 2 - 1
[6,2]
=> [[1,2,3,4,5,6],[7,8]]
=> [6,2] => [1,1] => 1 = 2 - 1
[6,1,1]
=> [[1,2,3,4,5,6],[7],[8]]
=> [6,1,1] => [1,2] => 2 = 3 - 1
[5,3]
=> [[1,2,3,4,5],[6,7,8]]
=> [5,3] => [1,1] => 1 = 2 - 1
[5,2,1]
=> [[1,2,3,4,5],[6,7],[8]]
=> [5,2,1] => [1,1,1] => 1 = 2 - 1
[1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10]]
=> [1,1,1,1,1,1,1,1,1,1] => [10] => ? = 2 - 1
Description
The number of different parts of an integer composition.
Matching statistic: St000318
Mp00095: Integer partitions to binary wordBinary words
Mp00178: Binary words to compositionInteger compositions
Mp00040: Integer compositions to partitionInteger partitions
St000318: Integer partitions ⟶ ℤResult quality: 82% values known / values provided: 82%distinct values known / distinct values provided: 100%
Values
[1]
=> 10 => [1,2] => [2,1]
=> 3 = 2 + 1
[2]
=> 100 => [1,3] => [3,1]
=> 3 = 2 + 1
[1,1]
=> 110 => [1,1,2] => [2,1,1]
=> 3 = 2 + 1
[3]
=> 1000 => [1,4] => [4,1]
=> 3 = 2 + 1
[2,1]
=> 1010 => [1,2,2] => [2,2,1]
=> 3 = 2 + 1
[1,1,1]
=> 1110 => [1,1,1,2] => [2,1,1,1]
=> 3 = 2 + 1
[4]
=> 10000 => [1,5] => [5,1]
=> 3 = 2 + 1
[3,1]
=> 10010 => [1,3,2] => [3,2,1]
=> 4 = 3 + 1
[2,2]
=> 1100 => [1,1,3] => [3,1,1]
=> 3 = 2 + 1
[2,1,1]
=> 10110 => [1,2,1,2] => [2,2,1,1]
=> 3 = 2 + 1
[1,1,1,1]
=> 11110 => [1,1,1,1,2] => [2,1,1,1,1]
=> 3 = 2 + 1
[5]
=> 100000 => [1,6] => [6,1]
=> 3 = 2 + 1
[4,1]
=> 100010 => [1,4,2] => [4,2,1]
=> 4 = 3 + 1
[3,2]
=> 10100 => [1,2,3] => [3,2,1]
=> 4 = 3 + 1
[3,1,1]
=> 100110 => [1,3,1,2] => [3,2,1,1]
=> 4 = 3 + 1
[2,2,1]
=> 11010 => [1,1,2,2] => [2,2,1,1]
=> 3 = 2 + 1
[2,1,1,1]
=> 101110 => [1,2,1,1,2] => [2,2,1,1,1]
=> 3 = 2 + 1
[1,1,1,1,1]
=> 111110 => [1,1,1,1,1,2] => [2,1,1,1,1,1]
=> 3 = 2 + 1
[6]
=> 1000000 => [1,7] => [7,1]
=> 3 = 2 + 1
[5,1]
=> 1000010 => [1,5,2] => [5,2,1]
=> 4 = 3 + 1
[4,2]
=> 100100 => [1,3,3] => [3,3,1]
=> 3 = 2 + 1
[4,1,1]
=> 1000110 => [1,4,1,2] => [4,2,1,1]
=> 4 = 3 + 1
[3,3]
=> 11000 => [1,1,4] => [4,1,1]
=> 3 = 2 + 1
[3,2,1]
=> 101010 => [1,2,2,2] => [2,2,2,1]
=> 3 = 2 + 1
[3,1,1,1]
=> 1001110 => [1,3,1,1,2] => [3,2,1,1,1]
=> 4 = 3 + 1
[2,2,2]
=> 11100 => [1,1,1,3] => [3,1,1,1]
=> 3 = 2 + 1
[2,2,1,1]
=> 110110 => [1,1,2,1,2] => [2,2,1,1,1]
=> 3 = 2 + 1
[2,1,1,1,1]
=> 1011110 => [1,2,1,1,1,2] => [2,2,1,1,1,1]
=> 3 = 2 + 1
[1,1,1,1,1,1]
=> 1111110 => [1,1,1,1,1,1,2] => [2,1,1,1,1,1,1]
=> 3 = 2 + 1
[7]
=> 10000000 => [1,8] => [8,1]
=> 3 = 2 + 1
[6,1]
=> 10000010 => [1,6,2] => [6,2,1]
=> 4 = 3 + 1
[5,2]
=> 1000100 => [1,4,3] => [4,3,1]
=> 4 = 3 + 1
[5,1,1]
=> 10000110 => [1,5,1,2] => [5,2,1,1]
=> 4 = 3 + 1
[4,3]
=> 101000 => [1,2,4] => [4,2,1]
=> 4 = 3 + 1
[4,2,1]
=> 1001010 => [1,3,2,2] => [3,2,2,1]
=> 4 = 3 + 1
[4,1,1,1]
=> 10001110 => [1,4,1,1,2] => [4,2,1,1,1]
=> 4 = 3 + 1
[3,3,1]
=> 110010 => [1,1,3,2] => [3,2,1,1]
=> 4 = 3 + 1
[3,2,2]
=> 101100 => [1,2,1,3] => [3,2,1,1]
=> 4 = 3 + 1
[3,2,1,1]
=> 1010110 => [1,2,2,1,2] => [2,2,2,1,1]
=> 3 = 2 + 1
[3,1,1,1,1]
=> 10011110 => [1,3,1,1,1,2] => [3,2,1,1,1,1]
=> 4 = 3 + 1
[2,2,2,1]
=> 111010 => [1,1,1,2,2] => [2,2,1,1,1]
=> 3 = 2 + 1
[2,2,1,1,1]
=> 1101110 => [1,1,2,1,1,2] => [2,2,1,1,1,1]
=> 3 = 2 + 1
[2,1,1,1,1,1]
=> 10111110 => [1,2,1,1,1,1,2] => [2,2,1,1,1,1,1]
=> 3 = 2 + 1
[1,1,1,1,1,1,1]
=> 11111110 => [1,1,1,1,1,1,1,2] => [2,1,1,1,1,1,1,1]
=> 3 = 2 + 1
[8]
=> 100000000 => [1,9] => [9,1]
=> 3 = 2 + 1
[7,1]
=> 100000010 => [1,7,2] => [7,2,1]
=> 4 = 3 + 1
[6,2]
=> 10000100 => [1,5,3] => [5,3,1]
=> 4 = 3 + 1
[6,1,1]
=> 100000110 => [1,6,1,2] => [6,2,1,1]
=> 4 = 3 + 1
[5,3]
=> 1001000 => [1,3,4] => [4,3,1]
=> 4 = 3 + 1
[5,2,1]
=> 10001010 => [1,4,2,2] => [4,2,2,1]
=> 4 = 3 + 1
[9]
=> 1000000000 => [1,10] => [10,1]
=> ? ∊ {2,2,2,3,3,3,3,3,3} + 1
[8,1]
=> 1000000010 => [1,8,2] => [8,2,1]
=> ? ∊ {2,2,2,3,3,3,3,3,3} + 1
[7,1,1]
=> 1000000110 => [1,7,1,2] => [7,2,1,1]
=> ? ∊ {2,2,2,3,3,3,3,3,3} + 1
[6,1,1,1]
=> 1000001110 => [1,6,1,1,2] => [6,2,1,1,1]
=> ? ∊ {2,2,2,3,3,3,3,3,3} + 1
[5,1,1,1,1]
=> 1000011110 => [1,5,1,1,1,2] => [5,2,1,1,1,1]
=> ? ∊ {2,2,2,3,3,3,3,3,3} + 1
[4,1,1,1,1,1]
=> 1000111110 => [1,4,1,1,1,1,2] => [4,2,1,1,1,1,1]
=> ? ∊ {2,2,2,3,3,3,3,3,3} + 1
[3,1,1,1,1,1,1]
=> 1001111110 => [1,3,1,1,1,1,1,2] => [3,2,1,1,1,1,1,1]
=> ? ∊ {2,2,2,3,3,3,3,3,3} + 1
[2,1,1,1,1,1,1,1]
=> 1011111110 => [1,2,1,1,1,1,1,1,2] => [2,2,1,1,1,1,1,1,1]
=> ? ∊ {2,2,2,3,3,3,3,3,3} + 1
[1,1,1,1,1,1,1,1,1]
=> 1111111110 => [1,1,1,1,1,1,1,1,1,2] => [2,1,1,1,1,1,1,1,1,1]
=> ? ∊ {2,2,2,3,3,3,3,3,3} + 1
[10]
=> 10000000000 => [1,11] => [11,1]
=> ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3} + 1
[9,1]
=> 10000000010 => [1,9,2] => [9,2,1]
=> ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3} + 1
[8,2]
=> 1000000100 => [1,7,3] => [7,3,1]
=> ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3} + 1
[8,1,1]
=> 10000000110 => [1,8,1,2] => [8,2,1,1]
=> ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3} + 1
[7,2,1]
=> 1000001010 => [1,6,2,2] => [6,2,2,1]
=> ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3} + 1
[7,1,1,1]
=> 10000001110 => [1,7,1,1,2] => [7,2,1,1,1]
=> ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3} + 1
[6,1,1,1,1]
=> 10000011110 => [1,6,1,1,1,2] => [6,2,1,1,1,1]
=> ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3} + 1
[5,2,1,1,1]
=> 1000101110 => [1,4,2,1,1,2] => [4,2,2,1,1,1]
=> ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3} + 1
[5,1,1,1,1,1]
=> 10000111110 => [1,5,1,1,1,1,2] => [5,2,1,1,1,1,1]
=> ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3} + 1
[4,2,1,1,1,1]
=> 1001011110 => [1,3,2,1,1,1,2] => [3,2,2,1,1,1,1]
=> ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3} + 1
[4,1,1,1,1,1,1]
=> 10001111110 => [1,4,1,1,1,1,1,2] => [4,2,1,1,1,1,1,1]
=> ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3} + 1
[3,2,1,1,1,1,1]
=> 1010111110 => [1,2,2,1,1,1,1,2] => [2,2,2,1,1,1,1,1]
=> ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3} + 1
[3,1,1,1,1,1,1,1]
=> 10011111110 => [1,3,1,1,1,1,1,1,2] => [3,2,1,1,1,1,1,1,1]
=> ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3} + 1
[2,2,1,1,1,1,1,1]
=> 1101111110 => [1,1,2,1,1,1,1,1,2] => [2,2,1,1,1,1,1,1,1]
=> ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3} + 1
[2,1,1,1,1,1,1,1,1]
=> 10111111110 => [1,2,1,1,1,1,1,1,1,2] => [2,2,1,1,1,1,1,1,1,1]
=> ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3} + 1
[1,1,1,1,1,1,1,1,1,1]
=> 11111111110 => [1,1,1,1,1,1,1,1,1,1,2] => [2,1,1,1,1,1,1,1,1,1,1]
=> ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3} + 1
Description
The number of addable cells of the Ferrers diagram of an integer partition.
Matching statistic: St001124
Mp00095: Integer partitions to binary wordBinary words
Mp00178: Binary words to compositionInteger compositions
Mp00040: Integer compositions to partitionInteger partitions
St001124: Integer partitions ⟶ ℤResult quality: 82% values known / values provided: 82%distinct values known / distinct values provided: 100%
Values
[1]
=> 10 => [1,2] => [2,1]
=> 1 = 2 - 1
[2]
=> 100 => [1,3] => [3,1]
=> 1 = 2 - 1
[1,1]
=> 110 => [1,1,2] => [2,1,1]
=> 1 = 2 - 1
[3]
=> 1000 => [1,4] => [4,1]
=> 1 = 2 - 1
[2,1]
=> 1010 => [1,2,2] => [2,2,1]
=> 1 = 2 - 1
[1,1,1]
=> 1110 => [1,1,1,2] => [2,1,1,1]
=> 1 = 2 - 1
[4]
=> 10000 => [1,5] => [5,1]
=> 1 = 2 - 1
[3,1]
=> 10010 => [1,3,2] => [3,2,1]
=> 2 = 3 - 1
[2,2]
=> 1100 => [1,1,3] => [3,1,1]
=> 1 = 2 - 1
[2,1,1]
=> 10110 => [1,2,1,2] => [2,2,1,1]
=> 1 = 2 - 1
[1,1,1,1]
=> 11110 => [1,1,1,1,2] => [2,1,1,1,1]
=> 1 = 2 - 1
[5]
=> 100000 => [1,6] => [6,1]
=> 1 = 2 - 1
[4,1]
=> 100010 => [1,4,2] => [4,2,1]
=> 2 = 3 - 1
[3,2]
=> 10100 => [1,2,3] => [3,2,1]
=> 2 = 3 - 1
[3,1,1]
=> 100110 => [1,3,1,2] => [3,2,1,1]
=> 2 = 3 - 1
[2,2,1]
=> 11010 => [1,1,2,2] => [2,2,1,1]
=> 1 = 2 - 1
[2,1,1,1]
=> 101110 => [1,2,1,1,2] => [2,2,1,1,1]
=> 1 = 2 - 1
[1,1,1,1,1]
=> 111110 => [1,1,1,1,1,2] => [2,1,1,1,1,1]
=> 1 = 2 - 1
[6]
=> 1000000 => [1,7] => [7,1]
=> 1 = 2 - 1
[5,1]
=> 1000010 => [1,5,2] => [5,2,1]
=> 2 = 3 - 1
[4,2]
=> 100100 => [1,3,3] => [3,3,1]
=> 1 = 2 - 1
[4,1,1]
=> 1000110 => [1,4,1,2] => [4,2,1,1]
=> 2 = 3 - 1
[3,3]
=> 11000 => [1,1,4] => [4,1,1]
=> 1 = 2 - 1
[3,2,1]
=> 101010 => [1,2,2,2] => [2,2,2,1]
=> 1 = 2 - 1
[3,1,1,1]
=> 1001110 => [1,3,1,1,2] => [3,2,1,1,1]
=> 2 = 3 - 1
[2,2,2]
=> 11100 => [1,1,1,3] => [3,1,1,1]
=> 1 = 2 - 1
[2,2,1,1]
=> 110110 => [1,1,2,1,2] => [2,2,1,1,1]
=> 1 = 2 - 1
[2,1,1,1,1]
=> 1011110 => [1,2,1,1,1,2] => [2,2,1,1,1,1]
=> 1 = 2 - 1
[1,1,1,1,1,1]
=> 1111110 => [1,1,1,1,1,1,2] => [2,1,1,1,1,1,1]
=> 1 = 2 - 1
[7]
=> 10000000 => [1,8] => [8,1]
=> 1 = 2 - 1
[6,1]
=> 10000010 => [1,6,2] => [6,2,1]
=> 2 = 3 - 1
[5,2]
=> 1000100 => [1,4,3] => [4,3,1]
=> 2 = 3 - 1
[5,1,1]
=> 10000110 => [1,5,1,2] => [5,2,1,1]
=> 2 = 3 - 1
[4,3]
=> 101000 => [1,2,4] => [4,2,1]
=> 2 = 3 - 1
[4,2,1]
=> 1001010 => [1,3,2,2] => [3,2,2,1]
=> 2 = 3 - 1
[4,1,1,1]
=> 10001110 => [1,4,1,1,2] => [4,2,1,1,1]
=> 2 = 3 - 1
[3,3,1]
=> 110010 => [1,1,3,2] => [3,2,1,1]
=> 2 = 3 - 1
[3,2,2]
=> 101100 => [1,2,1,3] => [3,2,1,1]
=> 2 = 3 - 1
[3,2,1,1]
=> 1010110 => [1,2,2,1,2] => [2,2,2,1,1]
=> 1 = 2 - 1
[3,1,1,1,1]
=> 10011110 => [1,3,1,1,1,2] => [3,2,1,1,1,1]
=> 2 = 3 - 1
[2,2,2,1]
=> 111010 => [1,1,1,2,2] => [2,2,1,1,1]
=> 1 = 2 - 1
[2,2,1,1,1]
=> 1101110 => [1,1,2,1,1,2] => [2,2,1,1,1,1]
=> 1 = 2 - 1
[2,1,1,1,1,1]
=> 10111110 => [1,2,1,1,1,1,2] => [2,2,1,1,1,1,1]
=> 1 = 2 - 1
[1,1,1,1,1,1,1]
=> 11111110 => [1,1,1,1,1,1,1,2] => [2,1,1,1,1,1,1,1]
=> 1 = 2 - 1
[8]
=> 100000000 => [1,9] => [9,1]
=> 1 = 2 - 1
[7,1]
=> 100000010 => [1,7,2] => [7,2,1]
=> 2 = 3 - 1
[6,2]
=> 10000100 => [1,5,3] => [5,3,1]
=> 2 = 3 - 1
[6,1,1]
=> 100000110 => [1,6,1,2] => [6,2,1,1]
=> 2 = 3 - 1
[5,3]
=> 1001000 => [1,3,4] => [4,3,1]
=> 2 = 3 - 1
[5,2,1]
=> 10001010 => [1,4,2,2] => [4,2,2,1]
=> 2 = 3 - 1
[9]
=> 1000000000 => [1,10] => [10,1]
=> ? ∊ {2,2,2,3,3,3,3,3,3} - 1
[8,1]
=> 1000000010 => [1,8,2] => [8,2,1]
=> ? ∊ {2,2,2,3,3,3,3,3,3} - 1
[7,1,1]
=> 1000000110 => [1,7,1,2] => [7,2,1,1]
=> ? ∊ {2,2,2,3,3,3,3,3,3} - 1
[6,1,1,1]
=> 1000001110 => [1,6,1,1,2] => [6,2,1,1,1]
=> ? ∊ {2,2,2,3,3,3,3,3,3} - 1
[5,1,1,1,1]
=> 1000011110 => [1,5,1,1,1,2] => [5,2,1,1,1,1]
=> ? ∊ {2,2,2,3,3,3,3,3,3} - 1
[4,1,1,1,1,1]
=> 1000111110 => [1,4,1,1,1,1,2] => [4,2,1,1,1,1,1]
=> ? ∊ {2,2,2,3,3,3,3,3,3} - 1
[3,1,1,1,1,1,1]
=> 1001111110 => [1,3,1,1,1,1,1,2] => [3,2,1,1,1,1,1,1]
=> ? ∊ {2,2,2,3,3,3,3,3,3} - 1
[2,1,1,1,1,1,1,1]
=> 1011111110 => [1,2,1,1,1,1,1,1,2] => [2,2,1,1,1,1,1,1,1]
=> ? ∊ {2,2,2,3,3,3,3,3,3} - 1
[1,1,1,1,1,1,1,1,1]
=> 1111111110 => [1,1,1,1,1,1,1,1,1,2] => [2,1,1,1,1,1,1,1,1,1]
=> ? ∊ {2,2,2,3,3,3,3,3,3} - 1
[10]
=> 10000000000 => [1,11] => [11,1]
=> ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3} - 1
[9,1]
=> 10000000010 => [1,9,2] => [9,2,1]
=> ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3} - 1
[8,2]
=> 1000000100 => [1,7,3] => [7,3,1]
=> ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3} - 1
[8,1,1]
=> 10000000110 => [1,8,1,2] => [8,2,1,1]
=> ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3} - 1
[7,2,1]
=> 1000001010 => [1,6,2,2] => [6,2,2,1]
=> ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3} - 1
[7,1,1,1]
=> 10000001110 => [1,7,1,1,2] => [7,2,1,1,1]
=> ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3} - 1
[6,1,1,1,1]
=> 10000011110 => [1,6,1,1,1,2] => [6,2,1,1,1,1]
=> ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3} - 1
[5,2,1,1,1]
=> 1000101110 => [1,4,2,1,1,2] => [4,2,2,1,1,1]
=> ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3} - 1
[5,1,1,1,1,1]
=> 10000111110 => [1,5,1,1,1,1,2] => [5,2,1,1,1,1,1]
=> ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3} - 1
[4,2,1,1,1,1]
=> 1001011110 => [1,3,2,1,1,1,2] => [3,2,2,1,1,1,1]
=> ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3} - 1
[4,1,1,1,1,1,1]
=> 10001111110 => [1,4,1,1,1,1,1,2] => [4,2,1,1,1,1,1,1]
=> ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3} - 1
[3,2,1,1,1,1,1]
=> 1010111110 => [1,2,2,1,1,1,1,2] => [2,2,2,1,1,1,1,1]
=> ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3} - 1
[3,1,1,1,1,1,1,1]
=> 10011111110 => [1,3,1,1,1,1,1,1,2] => [3,2,1,1,1,1,1,1,1]
=> ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3} - 1
[2,2,1,1,1,1,1,1]
=> 1101111110 => [1,1,2,1,1,1,1,1,2] => [2,2,1,1,1,1,1,1,1]
=> ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3} - 1
[2,1,1,1,1,1,1,1,1]
=> 10111111110 => [1,2,1,1,1,1,1,1,1,2] => [2,2,1,1,1,1,1,1,1,1]
=> ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3} - 1
[1,1,1,1,1,1,1,1,1,1]
=> 11111111110 => [1,1,1,1,1,1,1,1,1,1,2] => [2,1,1,1,1,1,1,1,1,1,1]
=> ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3} - 1
Description
The multiplicity of the standard representation in the Kronecker square corresponding to a partition. The Kronecker coefficient is the multiplicity $g_{\mu,\nu}^\lambda$ of the Specht module $S^\lambda$ in $S^\mu\otimes S^\nu$: $$ S^\mu\otimes S^\nu = \bigoplus_\lambda g_{\mu,\nu}^\lambda S^\lambda $$ This statistic records the Kronecker coefficient $g_{\lambda,\lambda}^{(n-1)1}$, for $\lambda\vdash n > 1$. For $n\leq1$ the statistic is undefined. It follows from [3, Prop.4.1] (or, slightly easier from [3, Thm.4.2]) that this is one less than [[St000159]], the number of distinct parts of the partition.
Mp00042: Integer partitions initial tableauStandard tableaux
Mp00207: Standard tableaux horizontal strip sizesInteger compositions
St000905: Integer compositions ⟶ ℤResult quality: 67% values known / values provided: 70%distinct values known / distinct values provided: 67%
Values
[1]
=> [[1]]
=> [1] => 1 = 2 - 1
[2]
=> [[1,2]]
=> [2] => 1 = 2 - 1
[1,1]
=> [[1],[2]]
=> [1,1] => 1 = 2 - 1
[3]
=> [[1,2,3]]
=> [3] => 1 = 2 - 1
[2,1]
=> [[1,2],[3]]
=> [2,1] => 1 = 2 - 1
[1,1,1]
=> [[1],[2],[3]]
=> [1,1,1] => 1 = 2 - 1
[4]
=> [[1,2,3,4]]
=> [4] => 1 = 2 - 1
[3,1]
=> [[1,2,3],[4]]
=> [3,1] => 1 = 2 - 1
[2,2]
=> [[1,2],[3,4]]
=> [2,2] => 1 = 2 - 1
[2,1,1]
=> [[1,2],[3],[4]]
=> [2,1,1] => 2 = 3 - 1
[1,1,1,1]
=> [[1],[2],[3],[4]]
=> [1,1,1,1] => 1 = 2 - 1
[5]
=> [[1,2,3,4,5]]
=> [5] => 1 = 2 - 1
[4,1]
=> [[1,2,3,4],[5]]
=> [4,1] => 1 = 2 - 1
[3,2]
=> [[1,2,3],[4,5]]
=> [3,2] => 1 = 2 - 1
[3,1,1]
=> [[1,2,3],[4],[5]]
=> [3,1,1] => 2 = 3 - 1
[2,2,1]
=> [[1,2],[3,4],[5]]
=> [2,2,1] => 2 = 3 - 1
[2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [2,1,1,1] => 2 = 3 - 1
[1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [1,1,1,1,1] => 1 = 2 - 1
[6]
=> [[1,2,3,4,5,6]]
=> [6] => 1 = 2 - 1
[5,1]
=> [[1,2,3,4,5],[6]]
=> [5,1] => 1 = 2 - 1
[4,2]
=> [[1,2,3,4],[5,6]]
=> [4,2] => 1 = 2 - 1
[4,1,1]
=> [[1,2,3,4],[5],[6]]
=> [4,1,1] => 2 = 3 - 1
[3,3]
=> [[1,2,3],[4,5,6]]
=> [3,3] => 1 = 2 - 1
[3,2,1]
=> [[1,2,3],[4,5],[6]]
=> [3,2,1] => 1 = 2 - 1
[3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> [3,1,1,1] => 2 = 3 - 1
[2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [2,2,2] => 1 = 2 - 1
[2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> [2,2,1,1] => 1 = 2 - 1
[2,1,1,1,1]
=> [[1,2],[3],[4],[5],[6]]
=> [2,1,1,1,1] => 2 = 3 - 1
[1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> [1,1,1,1,1,1] => 1 = 2 - 1
[7]
=> [[1,2,3,4,5,6,7]]
=> [7] => 1 = 2 - 1
[6,1]
=> [[1,2,3,4,5,6],[7]]
=> [6,1] => 1 = 2 - 1
[5,2]
=> [[1,2,3,4,5],[6,7]]
=> [5,2] => 1 = 2 - 1
[5,1,1]
=> [[1,2,3,4,5],[6],[7]]
=> [5,1,1] => 2 = 3 - 1
[4,3]
=> [[1,2,3,4],[5,6,7]]
=> [4,3] => 1 = 2 - 1
[4,2,1]
=> [[1,2,3,4],[5,6],[7]]
=> [4,2,1] => 1 = 2 - 1
[4,1,1,1]
=> [[1,2,3,4],[5],[6],[7]]
=> [4,1,1,1] => 2 = 3 - 1
[3,3,1]
=> [[1,2,3],[4,5,6],[7]]
=> [3,3,1] => 2 = 3 - 1
[3,2,2]
=> [[1,2,3],[4,5],[6,7]]
=> [3,2,2] => 2 = 3 - 1
[3,2,1,1]
=> [[1,2,3],[4,5],[6],[7]]
=> [3,2,1,1] => 2 = 3 - 1
[3,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7]]
=> [3,1,1,1,1] => 2 = 3 - 1
[2,2,2,1]
=> [[1,2],[3,4],[5,6],[7]]
=> [2,2,2,1] => 2 = 3 - 1
[2,2,1,1,1]
=> [[1,2],[3,4],[5],[6],[7]]
=> [2,2,1,1,1] => 2 = 3 - 1
[2,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7]]
=> [2,1,1,1,1,1] => 2 = 3 - 1
[1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> [1,1,1,1,1,1,1] => 1 = 2 - 1
[8]
=> [[1,2,3,4,5,6,7,8]]
=> [8] => 1 = 2 - 1
[7,1]
=> [[1,2,3,4,5,6,7],[8]]
=> [7,1] => 1 = 2 - 1
[6,2]
=> [[1,2,3,4,5,6],[7,8]]
=> [6,2] => 1 = 2 - 1
[6,1,1]
=> [[1,2,3,4,5,6],[7],[8]]
=> [6,1,1] => 2 = 3 - 1
[5,3]
=> [[1,2,3,4,5],[6,7,8]]
=> [5,3] => 1 = 2 - 1
[5,2,1]
=> [[1,2,3,4,5],[6,7],[8]]
=> [5,2,1] => 1 = 2 - 1
[10]
=> [[1,2,3,4,5,6,7,8,9,10]]
=> [10] => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4} - 1
[9,1]
=> [[1,2,3,4,5,6,7,8,9],[10]]
=> [9,1] => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4} - 1
[8,2]
=> [[1,2,3,4,5,6,7,8],[9,10]]
=> [8,2] => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4} - 1
[8,1,1]
=> [[1,2,3,4,5,6,7,8],[9],[10]]
=> [8,1,1] => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4} - 1
[7,3]
=> [[1,2,3,4,5,6,7],[8,9,10]]
=> [7,3] => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4} - 1
[7,2,1]
=> [[1,2,3,4,5,6,7],[8,9],[10]]
=> [7,2,1] => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4} - 1
[7,1,1,1]
=> [[1,2,3,4,5,6,7],[8],[9],[10]]
=> [7,1,1,1] => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4} - 1
[6,4]
=> [[1,2,3,4,5,6],[7,8,9,10]]
=> [6,4] => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4} - 1
[6,3,1]
=> [[1,2,3,4,5,6],[7,8,9],[10]]
=> [6,3,1] => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4} - 1
[6,2,2]
=> [[1,2,3,4,5,6],[7,8],[9,10]]
=> [6,2,2] => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4} - 1
[6,2,1,1]
=> [[1,2,3,4,5,6],[7,8],[9],[10]]
=> [6,2,1,1] => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4} - 1
[6,1,1,1,1]
=> [[1,2,3,4,5,6],[7],[8],[9],[10]]
=> [6,1,1,1,1] => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4} - 1
[5,5]
=> [[1,2,3,4,5],[6,7,8,9,10]]
=> [5,5] => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4} - 1
[5,4,1]
=> [[1,2,3,4,5],[6,7,8,9],[10]]
=> [5,4,1] => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4} - 1
[5,3,2]
=> [[1,2,3,4,5],[6,7,8],[9,10]]
=> [5,3,2] => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4} - 1
[5,3,1,1]
=> [[1,2,3,4,5],[6,7,8],[9],[10]]
=> [5,3,1,1] => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4} - 1
[5,2,2,1]
=> [[1,2,3,4,5],[6,7],[8,9],[10]]
=> [5,2,2,1] => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4} - 1
[5,2,1,1,1]
=> [[1,2,3,4,5],[6,7],[8],[9],[10]]
=> [5,2,1,1,1] => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4} - 1
[5,1,1,1,1,1]
=> [[1,2,3,4,5],[6],[7],[8],[9],[10]]
=> [5,1,1,1,1,1] => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4} - 1
[4,4,2]
=> [[1,2,3,4],[5,6,7,8],[9,10]]
=> [4,4,2] => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4} - 1
[4,4,1,1]
=> [[1,2,3,4],[5,6,7,8],[9],[10]]
=> [4,4,1,1] => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4} - 1
[4,3,3]
=> [[1,2,3,4],[5,6,7],[8,9,10]]
=> [4,3,3] => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4} - 1
[4,3,2,1]
=> [[1,2,3,4],[5,6,7],[8,9],[10]]
=> [4,3,2,1] => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4} - 1
[4,3,1,1,1]
=> [[1,2,3,4],[5,6,7],[8],[9],[10]]
=> [4,3,1,1,1] => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4} - 1
[4,2,2,2]
=> [[1,2,3,4],[5,6],[7,8],[9,10]]
=> [4,2,2,2] => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4} - 1
[4,2,2,1,1]
=> [[1,2,3,4],[5,6],[7,8],[9],[10]]
=> [4,2,2,1,1] => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4} - 1
[4,2,1,1,1,1]
=> [[1,2,3,4],[5,6],[7],[8],[9],[10]]
=> [4,2,1,1,1,1] => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4} - 1
[4,1,1,1,1,1,1]
=> [[1,2,3,4],[5],[6],[7],[8],[9],[10]]
=> [4,1,1,1,1,1,1] => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4} - 1
[3,3,3,1]
=> [[1,2,3],[4,5,6],[7,8,9],[10]]
=> [3,3,3,1] => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4} - 1
[3,3,2,2]
=> [[1,2,3],[4,5,6],[7,8],[9,10]]
=> [3,3,2,2] => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4} - 1
[3,3,2,1,1]
=> [[1,2,3],[4,5,6],[7,8],[9],[10]]
=> [3,3,2,1,1] => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4} - 1
[3,3,1,1,1,1]
=> [[1,2,3],[4,5,6],[7],[8],[9],[10]]
=> [3,3,1,1,1,1] => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4} - 1
[3,2,2,2,1]
=> [[1,2,3],[4,5],[6,7],[8,9],[10]]
=> [3,2,2,2,1] => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4} - 1
[3,2,2,1,1,1]
=> [[1,2,3],[4,5],[6,7],[8],[9],[10]]
=> [3,2,2,1,1,1] => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4} - 1
[3,2,1,1,1,1,1]
=> [[1,2,3],[4,5],[6],[7],[8],[9],[10]]
=> [3,2,1,1,1,1,1] => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4} - 1
[3,1,1,1,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7],[8],[9],[10]]
=> [3,1,1,1,1,1,1,1] => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4} - 1
[2,2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8],[9,10]]
=> [2,2,2,2,2] => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4} - 1
[2,2,2,2,1,1]
=> [[1,2],[3,4],[5,6],[7,8],[9],[10]]
=> [2,2,2,2,1,1] => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4} - 1
[2,2,2,1,1,1,1]
=> [[1,2],[3,4],[5,6],[7],[8],[9],[10]]
=> [2,2,2,1,1,1,1] => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4} - 1
[2,2,1,1,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8],[9],[10]]
=> [2,2,1,1,1,1,1,1] => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4} - 1
[2,1,1,1,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7],[8],[9],[10]]
=> [2,1,1,1,1,1,1,1,1] => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4} - 1
[1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10]]
=> [1,1,1,1,1,1,1,1,1,1] => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4} - 1
Description
The number of different multiplicities of parts of an integer composition.
Mp00202: Integer partitions first row removalInteger partitions
Mp00043: Integer partitions to Dyck pathDyck paths
Mp00228: Dyck paths reflect parallelogram polyominoDyck paths
St001200: Dyck paths ⟶ ℤResult quality: 66% values known / values provided: 66%distinct values known / distinct values provided: 67%
Values
[1]
=> []
=> []
=> ?
=> ? = 2
[2]
=> []
=> []
=> ?
=> ? ∊ {2,2}
[1,1]
=> [1]
=> [1,0,1,0]
=> [1,1,0,0]
=> ? ∊ {2,2}
[3]
=> []
=> []
=> ?
=> ? ∊ {2,2}
[2,1]
=> [1]
=> [1,0,1,0]
=> [1,1,0,0]
=> ? ∊ {2,2}
[1,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 2
[4]
=> []
=> []
=> ?
=> ? ∊ {2,3}
[3,1]
=> [1]
=> [1,0,1,0]
=> [1,1,0,0]
=> ? ∊ {2,3}
[2,2]
=> [2]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 2
[2,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 2
[1,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,0,0,0]
=> 2
[5]
=> []
=> []
=> ?
=> ? ∊ {3,3}
[4,1]
=> [1]
=> [1,0,1,0]
=> [1,1,0,0]
=> ? ∊ {3,3}
[3,2]
=> [2]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 2
[3,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 2
[2,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> 2
[2,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,0,0,0]
=> 2
[1,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 3
[6]
=> []
=> []
=> ?
=> ? ∊ {2,2,2}
[5,1]
=> [1]
=> [1,0,1,0]
=> [1,1,0,0]
=> ? ∊ {2,2,2}
[4,2]
=> [2]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 2
[4,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 2
[3,3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 2
[3,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> 2
[3,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,0,0,0]
=> 2
[2,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 3
[2,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 3
[2,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 3
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> ? ∊ {2,2,2}
[7]
=> []
=> []
=> ?
=> ? ∊ {3,3,3,3}
[6,1]
=> [1]
=> [1,0,1,0]
=> [1,1,0,0]
=> ? ∊ {3,3,3,3}
[5,2]
=> [2]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 2
[5,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 2
[4,3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 2
[4,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> 2
[4,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,0,0,0]
=> 2
[3,3,1]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 3
[3,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 3
[3,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 3
[3,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 3
[2,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 3
[2,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 2
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> ? ∊ {3,3,3,3}
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> ? ∊ {3,3,3,3}
[8]
=> []
=> []
=> ?
=> ? ∊ {2,2,3,3,3,3}
[7,1]
=> [1]
=> [1,0,1,0]
=> [1,1,0,0]
=> ? ∊ {2,2,3,3,3,3}
[6,2]
=> [2]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 2
[6,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 2
[5,3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 2
[5,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> 2
[5,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,0,0,0]
=> 2
[4,4]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 3
[4,3,1]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 3
[4,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 3
[4,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 3
[4,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 3
[3,3,2]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 3
[3,3,1,1]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[3,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 3
[3,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 2
[3,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> ? ∊ {2,2,3,3,3,3}
[2,2,2,2]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> 3
[2,2,2,1,1]
=> [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 2
[2,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> ? ∊ {2,2,3,3,3,3}
[2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> ? ∊ {2,2,3,3,3,3}
[1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? ∊ {2,2,3,3,3,3}
[9]
=> []
=> []
=> ?
=> ? ∊ {2,2,3,3,3,3,3,3,3}
[8,1]
=> [1]
=> [1,0,1,0]
=> [1,1,0,0]
=> ? ∊ {2,2,3,3,3,3,3,3,3}
[7,2]
=> [2]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 2
[7,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 2
[6,3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 2
[6,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> 2
[6,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,0,0,0]
=> 2
[5,4]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 3
[4,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> ? ∊ {2,2,3,3,3,3,3,3,3}
[3,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> ? ∊ {2,2,3,3,3,3,3,3,3}
[3,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> ? ∊ {2,2,3,3,3,3,3,3,3}
[2,2,2,1,1,1]
=> [2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> ? ∊ {2,2,3,3,3,3,3,3,3}
[2,2,1,1,1,1,1]
=> [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {2,2,3,3,3,3,3,3,3}
[2,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? ∊ {2,2,3,3,3,3,3,3,3}
[1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> ? ∊ {2,2,3,3,3,3,3,3,3}
[10]
=> []
=> []
=> ?
=> ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4}
[9,1]
=> [1]
=> [1,0,1,0]
=> [1,1,0,0]
=> ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4}
[5,5]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4}
[5,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4}
[4,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4}
[4,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4}
[3,3,1,1,1,1]
=> [3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4}
[3,2,2,1,1,1]
=> [2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4}
[3,2,1,1,1,1,1]
=> [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4}
[3,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4}
[2,2,2,2,2]
=> [2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,1,0,1,0,0,0]
=> ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4}
[2,2,2,2,1,1]
=> [2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,0,1,1,1,0,1,0,0,0]
=> ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4}
[2,2,2,1,1,1,1]
=> [2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1,0,1,1,0,1,1,1,0,0,0,0,0]
=> ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4}
[2,2,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,1,0,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4}
[2,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4}
[1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4}
Description
The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$.
Matching statistic: St001114
Mp00043: Integer partitions to Dyck pathDyck paths
Mp00031: Dyck paths to 312-avoiding permutationPermutations
Mp00223: Permutations runsortPermutations
St001114: Permutations ⟶ ℤResult quality: 62% values known / values provided: 62%distinct values known / distinct values provided: 100%
Values
[1]
=> [1,0,1,0]
=> [1,2] => [1,2] => 0 = 2 - 2
[2]
=> [1,1,0,0,1,0]
=> [2,1,3] => [1,3,2] => 0 = 2 - 2
[1,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => [1,3,2] => 0 = 2 - 2
[3]
=> [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [1,4,2,3] => 0 = 2 - 2
[2,1]
=> [1,0,1,0,1,0]
=> [1,2,3] => [1,2,3] => 0 = 2 - 2
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [1,4,2,3] => 0 = 2 - 2
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => [1,5,2,3,4] => 0 = 2 - 2
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [1,4,2,3] => 0 = 2 - 2
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [1,4,2,3] => 0 = 2 - 2
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [1,3,4,2] => 1 = 3 - 2
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [1,5,2,3,4] => 0 = 2 - 2
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [5,4,3,2,1,6] => [1,6,2,3,4,5] => 0 = 2 - 2
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [3,4,2,1,5] => [1,5,2,3,4] => 0 = 2 - 2
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [1,3,4,2] => 1 = 3 - 2
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [1,3,2,4] => 0 = 2 - 2
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [1,2,4,3] => 1 = 3 - 2
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,3,2] => [1,4,5,2,3] => 1 = 3 - 2
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,6,5,4,3,2] => [1,6,2,3,4,5] => 0 = 2 - 2
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [6,5,4,3,2,1,7] => [1,7,2,3,4,5,6] => ? ∊ {2,3} - 2
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [4,5,3,2,1,6] => [1,6,2,3,4,5] => 0 = 2 - 2
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [3,2,4,1,5] => [1,5,2,4,3] => 0 = 2 - 2
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [1,5,2,4,3] => 0 = 2 - 2
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,5,4] => [1,5,2,3,4] => 0 = 2 - 2
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,2,3,4] => 0 = 2 - 2
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [1,4,2,3,5] => 0 = 2 - 2
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [1,5,2,3,4] => 0 = 2 - 2
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [1,3,5,2,4] => 1 = 3 - 2
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,5,6,4,3,2] => [1,5,6,2,3,4] => 1 = 3 - 2
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,7,6,5,4,3,2] => [1,7,2,3,4,5,6] => ? ∊ {2,3} - 2
[7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [7,6,5,4,3,2,1,8] => [1,8,2,3,4,5,6,7] => ? ∊ {2,3,3,3} - 2
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [5,6,4,3,2,1,7] => [1,7,2,3,4,5,6] => ? ∊ {2,3,3,3} - 2
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [4,3,5,2,1,6] => [1,6,2,3,5,4] => 1 = 3 - 2
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [3,5,4,2,1,6] => [1,6,2,3,5,4] => 1 = 3 - 2
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => [1,4,5,2,3] => 1 = 3 - 2
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [1,5,2,3,4] => 0 = 2 - 2
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [1,4,2,5,3] => 0 = 2 - 2
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [1,5,2,3,4] => 0 = 2 - 2
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [1,4,5,2,3] => 1 = 3 - 2
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [1,3,4,5,2] => 0 = 2 - 2
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,5,4,6,3,2] => [1,5,2,3,4,6] => 0 = 2 - 2
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [1,2,5,3,4] => 1 = 3 - 2
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,4,6,5,3,2] => [1,4,6,2,3,5] => 1 = 3 - 2
[2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,6,7,5,4,3,2] => [1,6,7,2,3,4,5] => ? ∊ {2,3,3,3} - 2
[1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,8,7,6,5,4,3,2] => [1,8,2,3,4,5,6,7] => ? ∊ {2,3,3,3} - 2
[8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [8,7,6,5,4,3,2,1,9] => [1,9,2,3,4,5,6,7,8] => ? ∊ {2,3,3,3,3,3,3,3} - 2
[7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [6,7,5,4,3,2,1,8] => [1,8,2,3,4,5,6,7] => ? ∊ {2,3,3,3,3,3,3,3} - 2
[6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [5,4,6,3,2,1,7] => [1,7,2,3,4,6,5] => ? ∊ {2,3,3,3,3,3,3,3} - 2
[6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [4,6,5,3,2,1,7] => [1,7,2,3,4,6,5] => ? ∊ {2,3,3,3,3,3,3,3} - 2
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [4,3,2,5,1,6] => [1,6,2,5,3,4] => 0 = 2 - 2
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [3,4,5,2,1,6] => [1,6,2,3,4,5] => 0 = 2 - 2
[5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [2,5,4,3,1,6] => [1,6,2,5,3,4] => 0 = 2 - 2
[4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [4,3,2,1,6,5] => [1,6,2,3,4,5] => 0 = 2 - 2
[4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [1,4,5,2,3] => 1 = 3 - 2
[4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [1,4,2,3,5] => 0 = 2 - 2
[4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [1,3,4,2,5] => 1 = 3 - 2
[4,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,5,4,3,6,2] => [1,5,2,3,6,4] => 1 = 3 - 2
[3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [1,3,5,2,4] => 1 = 3 - 2
[3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [1,3,2,5,4] => 0 = 2 - 2
[3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [1,2,4,5,3] => 0 = 2 - 2
[3,2,1,1,1]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,4,5,6,3,2] => [1,4,5,6,2,3] => 0 = 2 - 2
[3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,6,5,7,4,3,2] => [1,6,2,3,4,5,7] => ? ∊ {2,3,3,3,3,3,3,3} - 2
[2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,5,7,6,4,3,2] => [1,5,7,2,3,4,6] => ? ∊ {2,3,3,3,3,3,3,3} - 2
[2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,7,8,6,5,4,3,2] => [1,7,8,2,3,4,5,6] => ? ∊ {2,3,3,3,3,3,3,3} - 2
[1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,9,8,7,6,5,4,3,2] => [1,9,2,3,4,5,6,7,8] => ? ∊ {2,3,3,3,3,3,3,3} - 2
[9]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [9,8,7,6,5,4,3,2,1,10] => [1,10,2,3,4,5,6,7,8,9] => ? ∊ {2,2,3,3,3,3,3,3,3,3,3,3,3,3} - 2
[8,1]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
=> [7,8,6,5,4,3,2,1,9] => [1,9,2,3,4,5,6,7,8] => ? ∊ {2,2,3,3,3,3,3,3,3,3,3,3,3,3} - 2
[7,2]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
=> [6,5,7,4,3,2,1,8] => [1,8,2,3,4,5,7,6] => ? ∊ {2,2,3,3,3,3,3,3,3,3,3,3,3,3} - 2
[7,1,1]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0]
=> [5,7,6,4,3,2,1,8] => [1,8,2,3,4,5,7,6] => ? ∊ {2,2,3,3,3,3,3,3,3,3,3,3,3,3} - 2
[6,3]
=> [1,1,1,1,1,0,0,0,1,0,0,0,1,0]
=> [5,4,3,6,2,1,7] => [1,7,2,3,6,4,5] => ? ∊ {2,2,3,3,3,3,3,3,3,3,3,3,3,3} - 2
[6,2,1]
=> [1,1,1,1,0,1,0,1,0,0,0,0,1,0]
=> [4,5,6,3,2,1,7] => [1,7,2,3,4,5,6] => ? ∊ {2,2,3,3,3,3,3,3,3,3,3,3,3,3} - 2
[6,1,1,1]
=> [1,1,1,0,1,1,1,0,0,0,0,0,1,0]
=> [3,6,5,4,2,1,7] => [1,7,2,3,6,4,5] => ? ∊ {2,2,3,3,3,3,3,3,3,3,3,3,3,3} - 2
[4,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,1,0,0,0]
=> [1,6,5,4,7,3,2] => [1,6,2,3,4,7,5] => ? ∊ {2,2,3,3,3,3,3,3,3,3,3,3,3,3} - 2
[3,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,5,6,7,4,3,2] => [1,5,6,7,2,3,4] => ? ∊ {2,2,3,3,3,3,3,3,3,3,3,3,3,3} - 2
[3,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [1,7,6,8,5,4,3,2] => [1,7,2,3,4,5,6,8] => ? ∊ {2,2,3,3,3,3,3,3,3,3,3,3,3,3} - 2
[2,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,4,7,6,5,3,2] => [1,4,7,2,3,5,6] => ? ∊ {2,2,3,3,3,3,3,3,3,3,3,3,3,3} - 2
[2,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [1,6,8,7,5,4,3,2] => [1,6,8,2,3,4,5,7] => ? ∊ {2,2,3,3,3,3,3,3,3,3,3,3,3,3} - 2
[2,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [1,8,9,7,6,5,4,3,2] => [1,8,9,2,3,4,5,6,7] => ? ∊ {2,2,3,3,3,3,3,3,3,3,3,3,3,3} - 2
[1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,10,9,8,7,6,5,4,3,2] => [1,10,2,3,4,5,6,7,8,9] => ? ∊ {2,2,3,3,3,3,3,3,3,3,3,3,3,3} - 2
[10]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> [10,9,8,7,6,5,4,3,2,1,11] => [1,11,2,3,4,5,6,7,8,9,10] => ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3} - 2
[9,1]
=> [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,1,0]
=> [8,9,7,6,5,4,3,2,1,10] => [1,10,2,3,4,5,6,7,8,9] => ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3} - 2
[8,2]
=> [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,1,0]
=> [7,6,8,5,4,3,2,1,9] => [1,9,2,3,4,5,6,8,7] => ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3} - 2
[8,1,1]
=> [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,1,0]
=> [6,8,7,5,4,3,2,1,9] => [1,9,2,3,4,5,6,8,7] => ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3} - 2
[7,3]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
=> [6,5,4,7,3,2,1,8] => [1,8,2,3,4,7,5,6] => ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3} - 2
[7,2,1]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,0,1,0]
=> [5,6,7,4,3,2,1,8] => [1,8,2,3,4,5,6,7] => ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3} - 2
[7,1,1,1]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,0,1,0]
=> [4,7,6,5,3,2,1,8] => [1,8,2,3,4,7,5,6] => ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3} - 2
[6,4]
=> [1,1,1,1,1,0,0,0,0,1,0,0,1,0]
=> [5,4,3,2,6,1,7] => [1,7,2,6,3,4,5] => ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3} - 2
[6,3,1]
=> [1,1,1,1,0,1,0,0,1,0,0,0,1,0]
=> [4,5,3,6,2,1,7] => [1,7,2,3,6,4,5] => ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3} - 2
[6,2,2]
=> [1,1,1,1,0,0,1,1,0,0,0,0,1,0]
=> [4,3,6,5,2,1,7] => [1,7,2,3,6,4,5] => ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3} - 2
[6,2,1,1]
=> [1,1,1,0,1,1,0,1,0,0,0,0,1,0]
=> [3,5,6,4,2,1,7] => [1,7,2,3,5,6,4] => ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3} - 2
[6,1,1,1,1]
=> [1,1,0,1,1,1,1,0,0,0,0,0,1,0]
=> [2,6,5,4,3,1,7] => [1,7,2,6,3,4,5] => ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3} - 2
[5,5]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [5,4,3,2,1,7,6] => [1,7,2,3,4,5,6] => ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3} - 2
[5,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,6,5,4,3,7,2] => [1,6,2,3,7,4,5] => ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3} - 2
[4,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,1,0,0,0]
=> [1,5,6,4,7,3,2] => [1,5,6,2,3,4,7] => ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3} - 2
[4,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [1,7,6,5,8,4,3,2] => [1,7,2,3,4,5,8,6] => ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3} - 2
[3,3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,5,4,7,6,3,2] => [1,5,2,3,4,7,6] => ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3} - 2
[3,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,4,6,7,5,3,2] => [1,4,6,7,2,3,5] => ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3} - 2
[3,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [1,6,7,8,5,4,3,2] => [1,6,7,8,2,3,4,5] => ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3} - 2
[3,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> [1,8,7,9,6,5,4,3,2] => [1,8,2,3,4,5,6,7,9] => ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3} - 2
[2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [2,1,7,6,5,4,3] => [1,7,2,3,4,5,6] => ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3} - 2
[2,2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [1,5,8,7,6,4,3,2] => [1,5,8,2,3,4,6,7] => ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3} - 2
Description
The number of odd descents of a permutation.
Mp00043: Integer partitions to Dyck pathDyck paths
Mp00327: Dyck paths inverse Kreweras complementDyck paths
Mp00118: Dyck paths swap returns and last descentDyck paths
St001165: Dyck paths ⟶ ℤResult quality: 61% values known / values provided: 61%distinct values known / distinct values provided: 67%
Values
[1]
=> [1,0,1,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2
[2]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 2
[1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> 2
[3]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 2
[2,1]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 2
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 2
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 2
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> 2
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 2
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 3
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 2
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> 2
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 2
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 2
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> 3
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 3
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0]
=> 3
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {2,2}
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> 2
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 2
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 2
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 2
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 3
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 2
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 3
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 2
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0,1,0]
=> 3
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> ? ∊ {2,2}
[7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? ∊ {2,3,3,3}
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> ? ∊ {2,3,3,3}
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,1,0,0,0]
=> 3
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> 2
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 2
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 2
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 3
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 2
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 3
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 2
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0,1,0]
=> 3
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 3
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0,1,0]
=> 3
[2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0,1,0]
=> ? ∊ {2,3,3,3}
[1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> ? ∊ {2,3,3,3}
[8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> ?
=> ?
=> ? ∊ {2,2,2,2,2,2,3,3}
[7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> ? ∊ {2,2,2,2,2,2,3,3}
[6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> ? ∊ {2,2,2,2,2,2,3,3}
[6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> ? ∊ {2,2,2,2,2,2,3,3}
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0,1,1,0,0]
=> 3
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,1,1,0,0,1,1,0,0,0]
=> 3
[5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> 2
[4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> 2
[4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 2
[4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 3
[4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 2
[4,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> 3
[3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3
[3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 3
[3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 3
[3,2,1,1,1]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> 3
[3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0,1,0]
=> ? ∊ {2,2,2,2,2,2,3,3}
[2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1,0,1,0,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0,1,0]
=> ? ∊ {2,2,2,2,2,2,3,3}
[2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,0,0,1,0]
=> ? ∊ {2,2,2,2,2,2,3,3}
[1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ?
=> ?
=> ? ∊ {2,2,2,2,2,2,3,3}
[9]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> ?
=> ?
=> ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,3,3}
[8,1]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
=> ?
=> ?
=> ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,3,3}
[7,2]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,1,1,0,1,0,1,1,0,0,0,0,0]
=> ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,3,3}
[7,1,1]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,3,3}
[6,3]
=> [1,1,1,1,1,0,0,0,1,0,0,0,1,0]
=> [1,0,1,1,0,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,1,1,0,0,0]
=> ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,3,3}
[6,2,1]
=> [1,1,1,1,0,1,0,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,1,1,0,0,0,0]
=> ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,3,3}
[6,1,1,1]
=> [1,1,1,0,1,1,1,0,0,0,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,3,3}
[4,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,1,0,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0,1,0]
=> ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,3,3}
[3,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0,1,0]
=> ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,3,3}
[3,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,1,0,0,0,1,0]
=> ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,3,3}
[2,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0,1,0]
=> ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,3,3}
[2,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,1,0,0,1,0]
=> ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,3,3}
[2,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> ?
=> ?
=> ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,3,3}
[1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> ?
=> ?
=> ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,3,3}
[10]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> ?
=> ?
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[9,1]
=> [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,1,0]
=> ?
=> ?
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[8,2]
=> [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,1,0]
=> ?
=> ?
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[8,1,1]
=> [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,1,0]
=> ?
=> ?
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[7,3]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,0,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,1,0,1,0,1,0,1,1,0,0,0,0]
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[7,2,1]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,1,1,1,1,0,0,1,1,0,0,0,0,0]
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[7,1,1,1]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[6,4]
=> [1,1,1,1,1,0,0,0,0,1,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,1,0,0]
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[6,3,1]
=> [1,1,1,1,0,1,0,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0,1,1,0,0]
=> [1,0,1,1,1,0,0,1,0,1,1,0,0,0]
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[6,2,2]
=> [1,1,1,1,0,0,1,1,0,0,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,1,1,0,0,0,0]
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[6,2,1,1]
=> [1,1,1,0,1,1,0,1,0,0,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,1,1,0,0,0,0]
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[6,1,1,1,1]
=> [1,1,0,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[5,5]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[5,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0,1,0]
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[4,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,1,0,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0,1,0]
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[4,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [1,1,0,1,0,1,1,0,1,0,1,0,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,1,0,0,0,1,0]
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[3,3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,1,0,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0,1,0]
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[3,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0,1,0]
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[3,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,1,0,0,0,0,1,0]
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[3,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> ?
=> ?
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[2,2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0,1,0]
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
Description
Number of simple modules with even projective dimension in the corresponding Nakayama algebra.
Mp00043: Integer partitions to Dyck pathDyck paths
Mp00129: Dyck paths to 321-avoiding permutation (Billey-Jockusch-Stanley)Permutations
Mp00068: Permutations Simion-Schmidt mapPermutations
St000872: Permutations ⟶ ℤResult quality: 61% values known / values provided: 61%distinct values known / distinct values provided: 67%
Values
[1]
=> [1,0,1,0]
=> [2,1] => [2,1] => 0 = 2 - 2
[2]
=> [1,1,0,0,1,0]
=> [1,3,2] => [1,3,2] => 0 = 2 - 2
[1,1]
=> [1,0,1,1,0,0]
=> [2,1,3] => [2,1,3] => 0 = 2 - 2
[3]
=> [1,1,1,0,0,0,1,0]
=> [1,2,4,3] => [1,4,3,2] => 0 = 2 - 2
[2,1]
=> [1,0,1,0,1,0]
=> [2,3,1] => [2,3,1] => 0 = 2 - 2
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,1,3,4] => [2,1,4,3] => 0 = 2 - 2
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,2,3,5,4] => [1,5,4,3,2] => 0 = 2 - 2
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [3,1,4,2] => [3,1,4,2] => 0 = 2 - 2
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,3,2,4] => [1,4,3,2] => 0 = 2 - 2
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [2,4,1,3] => [2,4,1,3] => 1 = 3 - 2
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => [2,1,5,4,3] => 0 = 2 - 2
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,2,3,4,6,5] => [1,6,5,4,3,2] => 0 = 2 - 2
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,1,2,5,3] => [4,1,5,3,2] => 1 = 3 - 2
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,3,4,2] => [1,4,3,2] => 0 = 2 - 2
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [2,1,4,3] => [2,1,4,3] => 0 = 2 - 2
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [2,3,1,4] => [2,4,1,3] => 1 = 3 - 2
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [2,5,1,3,4] => [2,5,1,4,3] => 1 = 3 - 2
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,1,3,4,5,6] => [2,1,6,5,4,3] => 0 = 2 - 2
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,2,3,4,5,7,6] => [1,7,6,5,4,3,2] => ? ∊ {2,2} - 2
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [5,1,2,3,6,4] => [5,1,6,4,3,2] => 1 = 3 - 2
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,4,2,5,3] => [1,5,4,3,2] => 0 = 2 - 2
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,5,4] => [3,1,5,4,2] => 0 = 2 - 2
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,2,4,3,5] => [1,5,4,3,2] => 0 = 2 - 2
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [2,3,4,1] => [2,4,3,1] => 0 = 2 - 2
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => [2,1,5,4,3] => 0 = 2 - 2
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => [1,5,4,3,2] => 0 = 2 - 2
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [2,4,1,3,5] => [2,5,1,4,3] => 1 = 3 - 2
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [2,6,1,3,4,5] => [2,6,1,5,4,3] => 1 = 3 - 2
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,1,3,4,5,6,7] => [2,1,7,6,5,4,3] => ? ∊ {2,2} - 2
[7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,2,3,4,5,6,8,7] => [1,8,7,6,5,4,3,2] => ? ∊ {3,3,3,3} - 2
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [6,1,2,3,4,7,5] => [6,1,7,5,4,3,2] => ? ∊ {3,3,3,3} - 2
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,5,2,3,6,4] => [1,6,5,4,3,2] => 0 = 2 - 2
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [4,1,2,3,6,5] => [4,1,6,5,3,2] => 1 = 3 - 2
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,2,4,5,3] => [1,5,4,3,2] => 0 = 2 - 2
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [3,4,1,5,2] => [3,5,1,4,2] => 1 = 3 - 2
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => [2,1,5,4,3] => 0 = 2 - 2
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [3,1,4,2,5] => [3,1,5,4,2] => 0 = 2 - 2
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,3,5,2,4] => [1,5,4,3,2] => 0 = 2 - 2
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => [2,5,4,1,3] => 1 = 3 - 2
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [2,1,6,3,4,5] => [2,1,6,5,4,3] => 0 = 2 - 2
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => [2,5,1,4,3] => 1 = 3 - 2
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [2,5,1,3,4,6] => [2,6,1,5,4,3] => 1 = 3 - 2
[2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [2,7,1,3,4,5,6] => [2,7,1,6,5,4,3] => ? ∊ {3,3,3,3} - 2
[1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [2,1,3,4,5,6,7,8] => [2,1,8,7,6,5,4,3] => ? ∊ {3,3,3,3} - 2
[8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,2,3,4,5,6,7,9,8] => [1,9,8,7,6,5,4,3,2] => ? ∊ {2,3,3,3,3,3,3,3} - 2
[7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [7,1,2,3,4,5,8,6] => [7,1,8,6,5,4,3,2] => ? ∊ {2,3,3,3,3,3,3,3} - 2
[6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [1,6,2,3,4,7,5] => [1,7,6,5,4,3,2] => ? ∊ {2,3,3,3,3,3,3,3} - 2
[6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [5,1,2,3,4,7,6] => [5,1,7,6,4,3,2] => ? ∊ {2,3,3,3,3,3,3,3} - 2
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,2,5,3,6,4] => [1,6,5,4,3,2] => 0 = 2 - 2
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [4,5,1,2,6,3] => [4,6,1,5,3,2] => 1 = 3 - 2
[5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [3,1,2,4,6,5] => [3,1,6,5,4,2] => 0 = 2 - 2
[4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,2,3,5,4,6] => [1,6,5,4,3,2] => 0 = 2 - 2
[4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> [3,1,4,5,2] => [3,1,5,4,2] => 0 = 2 - 2
[4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => [1,5,4,3,2] => 0 = 2 - 2
[4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0]
=> [2,4,1,5,3] => [2,5,1,4,3] => 1 = 3 - 2
[4,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [2,1,3,6,4,5] => [2,1,6,5,4,3] => 0 = 2 - 2
[3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => [1,5,4,3,2] => 0 = 2 - 2
[3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => [2,1,5,4,3] => 0 = 2 - 2
[3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [2,3,5,1,4] => [2,5,4,1,3] => 1 = 3 - 2
[3,2,1,1,1]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [2,5,6,1,3,4] => [2,6,5,1,4,3] => 1 = 3 - 2
[3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [2,1,7,3,4,5,6] => [2,1,7,6,5,4,3] => ? ∊ {2,3,3,3,3,3,3,3} - 2
[2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [2,6,1,3,4,5,7] => [2,7,1,6,5,4,3] => ? ∊ {2,3,3,3,3,3,3,3} - 2
[2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [2,8,1,3,4,5,6,7] => [2,8,1,7,6,5,4,3] => ? ∊ {2,3,3,3,3,3,3,3} - 2
[1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [2,1,3,4,5,6,7,8,9] => [2,1,9,8,7,6,5,4,3] => ? ∊ {2,3,3,3,3,3,3,3} - 2
[9]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [1,2,3,4,5,6,7,8,10,9] => [1,10,9,8,7,6,5,4,3,2] => ? ∊ {2,2,2,3,3,3,3,3,3,3,3,3,3,3} - 2
[8,1]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
=> [8,1,2,3,4,5,6,9,7] => [8,1,9,7,6,5,4,3,2] => ? ∊ {2,2,2,3,3,3,3,3,3,3,3,3,3,3} - 2
[7,2]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
=> [1,7,2,3,4,5,8,6] => [1,8,7,6,5,4,3,2] => ? ∊ {2,2,2,3,3,3,3,3,3,3,3,3,3,3} - 2
[7,1,1]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0]
=> [6,1,2,3,4,5,8,7] => [6,1,8,7,5,4,3,2] => ? ∊ {2,2,2,3,3,3,3,3,3,3,3,3,3,3} - 2
[6,3]
=> [1,1,1,1,1,0,0,0,1,0,0,0,1,0]
=> [1,2,6,3,4,7,5] => [1,7,6,5,4,3,2] => ? ∊ {2,2,2,3,3,3,3,3,3,3,3,3,3,3} - 2
[6,2,1]
=> [1,1,1,1,0,1,0,1,0,0,0,0,1,0]
=> [5,6,1,2,3,7,4] => [5,7,1,6,4,3,2] => ? ∊ {2,2,2,3,3,3,3,3,3,3,3,3,3,3} - 2
[6,1,1,1]
=> [1,1,1,0,1,1,1,0,0,0,0,0,1,0]
=> [4,1,2,3,5,7,6] => [4,1,7,6,5,3,2] => ? ∊ {2,2,2,3,3,3,3,3,3,3,3,3,3,3} - 2
[4,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,1,0,0,0]
=> [2,1,3,7,4,5,6] => [2,1,7,6,5,4,3] => ? ∊ {2,2,2,3,3,3,3,3,3,3,3,3,3,3} - 2
[3,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [2,6,7,1,3,4,5] => [2,7,6,1,5,4,3] => ? ∊ {2,2,2,3,3,3,3,3,3,3,3,3,3,3} - 2
[3,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [2,1,8,3,4,5,6,7] => [2,1,8,7,6,5,4,3] => ? ∊ {2,2,2,3,3,3,3,3,3,3,3,3,3,3} - 2
[2,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [2,5,1,3,4,6,7] => [2,7,1,6,5,4,3] => ? ∊ {2,2,2,3,3,3,3,3,3,3,3,3,3,3} - 2
[2,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [2,7,1,3,4,5,6,8] => [2,8,1,7,6,5,4,3] => ? ∊ {2,2,2,3,3,3,3,3,3,3,3,3,3,3} - 2
[2,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [2,9,1,3,4,5,6,7,8] => [2,9,1,8,7,6,5,4,3] => ? ∊ {2,2,2,3,3,3,3,3,3,3,3,3,3,3} - 2
[1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [2,1,3,4,5,6,7,8,9,10] => [2,1,10,9,8,7,6,5,4,3] => ? ∊ {2,2,2,3,3,3,3,3,3,3,3,3,3,3} - 2
[10]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> [1,2,3,4,5,6,7,8,9,11,10] => [1,11,10,9,8,7,6,5,4,3,2] => ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4} - 2
[9,1]
=> [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,1,0]
=> [9,1,2,3,4,5,6,7,10,8] => [9,1,10,8,7,6,5,4,3,2] => ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4} - 2
[8,2]
=> [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,1,0]
=> [1,8,2,3,4,5,6,9,7] => [1,9,8,7,6,5,4,3,2] => ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4} - 2
[8,1,1]
=> [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,1,0]
=> [7,1,2,3,4,5,6,9,8] => [7,1,9,8,6,5,4,3,2] => ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4} - 2
[7,3]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
=> [1,2,7,3,4,5,8,6] => [1,8,7,6,5,4,3,2] => ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4} - 2
[7,2,1]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,0,1,0]
=> [6,7,1,2,3,4,8,5] => [6,8,1,7,5,4,3,2] => ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4} - 2
[7,1,1,1]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,0,1,0]
=> [5,1,2,3,4,6,8,7] => [5,1,8,7,6,4,3,2] => ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4} - 2
[6,4]
=> [1,1,1,1,1,0,0,0,0,1,0,0,1,0]
=> [1,2,3,6,4,7,5] => [1,7,6,5,4,3,2] => ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4} - 2
[6,3,1]
=> [1,1,1,1,0,1,0,0,1,0,0,0,1,0]
=> [5,1,6,2,3,7,4] => [5,1,7,6,4,3,2] => ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4} - 2
[6,2,2]
=> [1,1,1,1,0,0,1,1,0,0,0,0,1,0]
=> [1,5,2,3,4,7,6] => [1,7,6,5,4,3,2] => ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4} - 2
[6,2,1,1]
=> [1,1,1,0,1,1,0,1,0,0,0,0,1,0]
=> [4,6,1,2,3,7,5] => [4,7,1,6,5,3,2] => ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4} - 2
[6,1,1,1,1]
=> [1,1,0,1,1,1,1,0,0,0,0,0,1,0]
=> [3,1,2,4,5,7,6] => [3,1,7,6,5,4,2] => ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4} - 2
[5,5]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [1,2,3,4,6,5,7] => [1,7,6,5,4,3,2] => ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4} - 2
[5,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [2,1,3,4,7,5,6] => [2,1,7,6,5,4,3] => ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4} - 2
[4,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,1,0,0,0]
=> [2,6,1,7,3,4,5] => [2,7,1,6,5,4,3] => ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4} - 2
[4,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [2,1,3,8,4,5,6,7] => [2,1,8,7,6,5,4,3] => ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4} - 2
[3,3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,1,0,0,0,0]
=> [2,1,6,3,4,5,7] => [2,1,7,6,5,4,3] => ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4} - 2
[3,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> [2,5,7,1,3,4,6] => [2,7,6,1,5,4,3] => ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4} - 2
[3,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [2,7,8,1,3,4,5,6] => [2,8,7,1,6,5,4,3] => ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4} - 2
[3,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> [2,1,9,3,4,5,6,7,8] => [2,1,9,8,7,6,5,4,3] => ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4} - 2
[2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [1,3,2,4,5,6,7] => [1,7,6,5,4,3,2] => ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4} - 2
[2,2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> [2,4,1,3,5,6,7] => [2,7,1,6,5,4,3] => ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4} - 2
Description
The number of very big descents of a permutation. A very big descent of a permutation $\pi$ is an index $i$ such that $\pi_i - \pi_{i+1} > 2$. For the number of descents, see [[St000021]] and for the number of big descents, see [[St000647]]. General $r$-descents were for example be studied in [1, Section 2].
The following 223 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001221The number of simple modules in the corresponding LNakayama algebra that have 2 dimensional second Extension group with the regular module. St001964The interval resolution global dimension of a poset. St001198The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001206The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St000966Number of peaks minus the global dimension of the corresponding LNakayama algebra. St000181The number of connected components of the Hasse diagram for the poset. St001890The maximum magnitude of the Möbius function of a poset. St000260The radius of a connected graph. St000470The number of runs in a permutation. St000353The number of inner valleys of a permutation. St000354The number of recoils of a permutation. St001469The holeyness of a permutation. St001729The number of visible descents of a permutation. St001737The number of descents of type 2 in a permutation. St001928The number of non-overlapping descents in a permutation. St001942The number of loops of the quiver corresponding to the reduced incidence algebra of a poset. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001665The number of pure excedances of a permutation. St001043The depth of the leaf closest to the root in the binary unordered tree associated with the perfect matching. St000007The number of saliances of the permutation. St000031The number of cycles in the cycle decomposition of a permutation. St000834The number of right outer peaks of a permutation. St001085The number of occurrences of the vincular pattern |21-3 in a permutation. St001394The genus of a permutation. St000908The length of the shortest maximal antichain in a poset. St000914The sum of the values of the Möbius function of a poset. St001532The leading coefficient of the Poincare polynomial of the poset cone. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St001301The first Betti number of the order complex associated with the poset. St001396Number of triples of incomparable elements in a finite poset. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St000035The number of left outer peaks of a permutation. St000352The Elizalde-Pak rank of a permutation. St000256The number of parts from which one can substract 2 and still get an integer partition. St000092The number of outer peaks of a permutation. St000542The number of left-to-right-minima of a permutation. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St001001The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001222Number of simple modules in the corresponding LNakayama algebra that have a unique 2-extension with the regular module. St001728The number of invisible descents of a permutation. St000061The number of nodes on the left branch of a binary tree. St000099The number of valleys of a permutation, including the boundary. St000314The number of left-to-right-maxima of a permutation. St000325The width of the tree associated to a permutation. St000955Number of times one has $Ext^i(D(A),A)>0$ for $i>0$ for the corresponding LNakayama algebra. St000991The number of right-to-left minima of a permutation. St001201The grade of the simple module $S_0$ in the special CNakayama algebra corresponding to the Dyck path. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St001741The largest integer such that all patterns of this size are contained in the permutation. St000021The number of descents of a permutation. St000023The number of inner peaks of a permutation. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000654The first descent of a permutation. St000711The number of big exceedences of a permutation. St000779The tier of a permutation. St000864The number of circled entries of the shifted recording tableau of a permutation. St000968We make a CNakayama algebra out of the LNakayama algebra (corresponding to the Dyck path) $[c_0,c_1,...,c_{n−1}]$ by adding $c_0$ to $c_{n−1}$. St001006Number of simple modules with projective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001162The minimum jump of a permutation. St001202Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n−1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. St001205The number of non-simple indecomposable projective-injective modules of the algebra $eAe$ in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001208The number of connected components of the quiver of $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra $A$ of $K[x]/(x^n)$. St001257The dominant dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St001344The neighbouring number of a permutation. St001493The number of simple modules with maximal even projective dimension in the corresponding Nakayama algebra. St001503The largest distance of a vertex to a vertex in a cycle in the resolution quiver of the corresponding Nakayama algebra. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001569The maximal modular displacement of a permutation. St001661Half the permanent of the Identity matrix plus the permutation matrix associated to the permutation. St001859The number of factors of the Stanley symmetric function associated with a permutation. St000252The number of nodes of degree 3 of a binary tree. St000375The number of non weak exceedences of a permutation that are mid-points of a decreasing subsequence of length $3$. St000664The number of right ropes of a permutation. St001059Number of occurrences of the patterns 41352,42351,51342,52341 in a permutation. St001163The number of simple modules with dominant dimension at least three in the corresponding Nakayama algebra. St001217The projective dimension of the indecomposable injective module I[n-2] in the corresponding Nakayama algebra with simples enumerated from 0 to n-1. St001230The number of simple modules with injective dimension equal to the dominant dimension equal to one and the dual property. St001264The smallest index i such that the i-th simple module has projective dimension equal to the global dimension of the corresponding Nakayama algebra. St001513The number of nested exceedences of a permutation. St001520The number of strict 3-descents. St001549The number of restricted non-inversions between exceedances. St001551The number of restricted non-inversions between exceedances where the rightmost exceedance is linked. St001715The number of non-records in a permutation. St001960The number of descents of a permutation minus one if its first entry is not one. St001060The distinguishing index of a graph. St000871The number of very big ascents of a permutation. St001095The number of non-isomorphic posets with precisely one further covering relation. St000058The order of a permutation. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000741The Colin de Verdière graph invariant. St000454The largest eigenvalue of a graph if it is integral. St001194The injective dimension of $A/AfA$ in the corresponding Nakayama algebra $A$ when $Af$ is the minimal faithful projective-injective left $A$-module St001231The number of simple modules that are non-projective and non-injective with the property that they have projective dimension equal to one and that also the Auslander-Reiten translates of the module and the inverse Auslander-Reiten translate of the module have the same projective dimension. St001234The number of indecomposable three dimensional modules with projective dimension one. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001490The number of connected components of a skew partition. St001720The minimal length of a chain of small intervals in a lattice. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St001846The number of elements which do not have a complement in the lattice. St001771The number of occurrences of the signed pattern 1-2 in a signed permutation. St001866The nesting alignments of a signed permutation. St001870The number of positive entries followed by a negative entry in a signed permutation. St001895The oddness of a signed permutation. St001330The hat guessing number of a graph. St001616The number of neutral elements in a lattice. St001613The binary logarithm of the size of the center of a lattice. St001881The number of factors of a lattice as a Cartesian product of lattices. St000326The position of the first one in a binary word after appending a 1 at the end. St000629The defect of a binary word. St001896The number of right descents of a signed permutations. St001487The number of inner corners of a skew partition. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. St000264The girth of a graph, which is not a tree. St001811The Castelnuovo-Mumford regularity of a permutation. St000842The breadth of a permutation. St000994The number of cycle peaks and the number of cycle valleys of a permutation. St000891The number of distinct diagonal sums of a permutation matrix. St000022The number of fixed points of a permutation. St000153The number of adjacent cycles of a permutation. St000405The number of occurrences of the pattern 1324 in a permutation. St001465The number of adjacent transpositions in the cycle decomposition of a permutation. St000877The depth of the binary word interpreted as a path. St000885The number of critical steps in the Catalan decomposition of a binary word. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St000876The number of factors in the Catalan decomposition of a binary word. St001256Number of simple reflexive modules that are 2-stable reflexive. St001722The number of minimal chains with small intervals between a binary word and the top element. St000297The number of leading ones in a binary word. St000455The second largest eigenvalue of a graph if it is integral. St000750The number of occurrences of the pattern 4213 in a permutation. St000751The number of occurrences of either of the pattern 2143 or 2143 in a permutation. St001140Number of indecomposable modules with projective and injective dimension at least two in the corresponding Nakayama algebra. St001171The vector space dimension of $Ext_A^1(I_o,A)$ when $I_o$ is the tilting module corresponding to the permutation $o$ in the Auslander algebra $A$ of $K[x]/(x^n)$. St001292The injective dimension of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001578The minimal number of edges to add or remove to make a graph a line graph. St000878The number of ones minus the number of zeros of a binary word. St000527The width of the poset. St000760The length of the longest strictly decreasing subsequence of parts of an integer composition. St000764The number of strong records in an integer composition. St000627The exponent of a binary word. St000807The sum of the heights of the valleys of the associated bargraph. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. St001371The length of the longest Yamanouchi prefix of a binary word. St001730The number of times the path corresponding to a binary word crosses the base line. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St000193The row of the unique '1' in the first column of the alternating sign matrix. St000392The length of the longest run of ones in a binary word. St000758The length of the longest staircase fitting into an integer composition. St000982The length of the longest constant subword. St001133The smallest label in the subtree rooted at the sister of 1 in the decreasing labelled binary unordered tree associated with the perfect matching. St001269The sum of the minimum of the number of exceedances and deficiencies in each cycle of a permutation. St001275The projective dimension of the second term in a minimal injective coresolution of the regular module. St001884The number of borders of a binary word. St000056The decomposition (or block) number of a permutation. St000295The length of the border of a binary word. St000296The length of the symmetric border of a binary word. St000317The cycle descent number of a permutation. St000657The smallest part of an integer composition. St000694The number of affine bounded permutations that project to a given permutation. St000761The number of ascents in an integer composition. St000767The number of runs in an integer composition. St000788The number of nesting-similar perfect matchings of a perfect matching. St000805The number of peaks of the associated bargraph. St000900The minimal number of repetitions of a part in an integer composition. St000902 The minimal number of repetitions of an integer composition. St001041The depth of the label 1 in the decreasing labelled binary unordered tree associated with the perfect matching. St001174The Gorenstein dimension of the algebra $A/I$ when $I$ is the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. St001260The permanent of an alternating sign matrix. St001267The length of the Lyndon factorization of the binary word. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001413Half the length of the longest even length palindromic prefix of a binary word. St001437The flex of a binary word. St001461The number of topologically connected components of the chord diagram of a permutation. St001462The number of factors of a standard tableaux under concatenation. St001481The minimal height of a peak of a Dyck path. St001507The sum of projective dimension of simple modules with even projective dimension divided by 2 in the LNakayama algebra corresponding to Dyck paths. St001566The length of the longest arithmetic progression in a permutation. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001590The crossing number of a perfect matching. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St001768The number of reduced words of a signed permutation. St001830The chord expansion number of a perfect matching. St001832The number of non-crossing perfect matchings in the chord expansion of a perfect matching. St001889The size of the connectivity set of a signed permutation. St001946The number of descents in a parking function. St000188The area of the Dyck path corresponding to a parking function and the total displacement of a parking function. St000195The number of secondary dinversion pairs of the dyck path corresponding to a parking function. St000221The number of strong fixed points of a permutation. St000279The size of the preimage of the map 'cycle-as-one-line notation' from Permutations to Permutations. St000407The number of occurrences of the pattern 2143 in a permutation. St000623The number of occurrences of the pattern 52341 in a permutation. St000666The number of right tethers of a permutation. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St000787The number of flips required to make a perfect matching noncrossing. St000894The trace of an alternating sign matrix. St000943The number of spots the most unlucky car had to go further in a parking function. St001131The number of trivial trees on the path to label one in the decreasing labelled binary unordered tree associated with the perfect matching. St001204Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n−1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St001381The fertility of a permutation. St001429The number of negative entries in a signed permutation. St001436The index of a given binary word in the lex-order among all its cyclic shifts. St001444The rank of the skew-symmetric form which is non-zero on crossing arcs of a perfect matching. St001466The number of transpositions swapping cyclically adjacent numbers in a permutation. St001524The degree of symmetry of a binary word. St001552The number of inversions between excedances and fixed points of a permutation. St001663The number of occurrences of the Hertzsprung pattern 132 in a permutation. St001810The number of fixed points of a permutation smaller than its largest moved point. St001831The multiplicity of the non-nesting perfect matching in the chord expansion of a perfect matching. St001837The number of occurrences of a 312 pattern in the restricted growth word of a perfect matching. St001845The number of join irreducibles minus the rank of a lattice. St001850The number of Hecke atoms of a permutation. St001856The number of edges in the reduced word graph of a permutation. St001867The number of alignments of type EN of a signed permutation. St001868The number of alignments of type NE of a signed permutation. St001473The absolute value of the sum of all entries of the Coxeter matrix of the corresponding LNakayama algebra. St001872The number of indecomposable injective modules with even projective dimension in the corresponding Nakayama algebra.